Detecting Homeomorphic 3-manifolds via Graph Neural Networks

TL;DR

This paper explores using Graph Neural Networks to efficiently determine whether two 3-manifolds are homeomorphic, offering a polynomial-time approximation to a traditionally super-polynomial problem.

Contribution

The study introduces a GNN-based approach for the homeomorphism problem of graph-manifolds, providing a novel machine learning perspective and benchmarking various architectures.

Findings

GNNs can classify homeomorphic graph-manifolds in polynomial time.

Different GNN architectures show varying strengths and weaknesses.

The dataset of plumbing graph pairs enables supervised learning for this topological problem.

Abstract

Motivated by the enumeration of the BPS spectra of certain 3d $\mathcal{N}=2$ supersymmetric quantum field theories, obtained from the compactification of 6d superconformal field theories on three-manifolds, we study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. Utilizing the JSJ decomposition, a unique representation via a plumbing graph is extracted from a graph-manifold. Homeomorphic graph-manifolds are related via a sequence of von Neumann moves on this graph; the algorithmic application of these moves can determine if two graphs correspond to homeomorphic graph-manifolds in super-polynomial time. However, by employing Graph Neural Networks (GNNs), the same problem can be addressed, at the cost of accuracy, in polynomial time. We build a dataset composed of pairs of plumbing graphs, together with a hidden label encoding whether the pair is homeomorphic. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers (GEN, GCN, GAT, NNConv), followed by an aggregation layer and a classification layer. We discuss the strengths and weaknesses of the different GNNs for this homeomorphism problem.